A stochastic look-ahead approach for hurricane relief logistics operations planning under uncertainty

Abstract

In the aftermath of a hurricane, humanitarian logistics plays a critical role in delivering relief items to the affected areas in a timely fashion. This paper proposes a novel stochastic look-ahead framework that implements a two-stage stochastic programming model in a rolling horizon approach to address the evolving uncertain logistics system state during the post-hurricane humanitarian logistics operations. The two-stage stochastic programming model that executes in this rolling horizon approach is formulated as a mixed-integer programming problem. The model aims to minimize the total cost incurred in the logistics operations, which consist of transportation cost and social cost. The social cost is measured as a function of deprivation for unsatisfied demand. Our extensive numerical results and sensitivity analysis demonstrate the effectiveness of the proposed approach in reducing the total cost incurred during the post-hurricane relief logistics operations compared to the two-stage stochastic programming model implemented in a static fashion.

Appendix: The static two-stage stochastic programming formulation for the post-hurricane relief logistics planning problem

$$\begin{aligned}&{\mathop {\min }} \quad z=\sum \limits _{t\in \mathcal {T}} \Big ( \sum \limits _{i\in \mathcal {L}}\eta _{i}y_{it} + \sum \limits _{i\in \mathcal {L}}\sum \limits _{k\in \mathcal {K}} \zeta _{k}V_{ikt} \Big ) \nonumber \\&\quad + \sum \limits _{\omega \in \Omega } P^{\omega } \sum \limits _{k\in \mathcal {K}}\sum \limits _{t\in \mathcal {T}} \Big (\sum \limits _{i\in L} B^{g}_{i} g^{\omega }_{ikt} + \sum \limits _{i\in \mathcal {L}\cup \mathcal {S}} B^{h}_{i} h^{\omega }_{ikt} + \sum \limits _{i\in \mathcal {L}\cup \mathcal {S}}\sum \limits _{j\in \mathcal {L}\cup \mathcal {S}, j\ne i} B_{ij} f^{\omega }_{ijkt}\Big )\nonumber \\&\quad +\sum \limits _{\omega \in \Omega } P^{\omega } \Big (\sum \limits _{i\in \mathcal {S}}\sum \limits _{k\in \mathcal {K}}\sum \limits _{t\in \mathcal {T}{\setminus }\{0\}}dep[w, i, k, U_{ik(t-1)}, t-1]\alpha _{ikt} \nonumber \\&\quad +\sum \limits _{i\in \mathcal {S}}\sum \limits _{k\in \mathcal {K}}dep[w, i, k, U_{ikT}, T]\Big ) \end{aligned}$$

(4a)

$$\begin{aligned}&\hbox {s.t.}\nonumber \\&V_{ik0}, U_{ik0}{= 0, \forall i\in \mathcal {S},k \in \mathcal {K}}{} \end{aligned}$$

(4b)

$$\begin{aligned}&V_{ik0}{= g^{\omega }_{ik0}, \quad \forall i\in \mathcal {L}, k \in \mathcal {K}, \omega \in \Omega }{} \end{aligned}$$

(4c)

$$\begin{aligned}&h^{\omega }_{ik0}{= 0, \quad \forall i\in \mathcal {L}\cup S, k \in \mathcal {K}, \omega \in \Omega }{} \end{aligned}$$

(4d)

$$\begin{aligned}&\sum \limits _{{\scriptscriptstyle j\in \mathcal {L} \cup \mathcal {S}, j \ne i }}f^{\omega }_{ijk0}{= 0, \quad \forall i \in \mathcal {L} \cup \mathcal {S}, k \in \mathcal {K}, \omega \in \Omega }{} \end{aligned}$$

(4e)

$$\begin{aligned}&x_{it}{= \sum ^{t}_{t^{'} = 0}y_{it^{'}}, \quad \forall i\in \mathcal {L}, t\in \mathcal {T}}{} \end{aligned}$$

(4f)

$$\begin{aligned}&{\sum _{t\in \mathcal {T}}y_{it}}{\le 1, \quad \forall i\in \mathcal {L}}{} \end{aligned}$$

(4g)

$$\begin{aligned}&{V_{ikt}}{\le \phi _{ik} x_{it}, \quad \forall i\in \mathcal {L}, k \in \mathcal {K}, t\in \mathcal {T}}{} \end{aligned}$$

(4h)

$$\begin{aligned}&{g^{\omega }_{ikt} + h^{\omega }_{ikt}}{\le V_{ikt}, \quad \forall i\in \mathcal {L}, k \in \mathcal {K}, t\in \mathcal {T}, \omega \in \Omega }{} \end{aligned}$$

(4i)

$$\begin{aligned}&{\sum \limits _{i\in \mathcal {L}}g^{\omega }_{ikt}}{\le R^{\omega }_{kt}, \quad \forall k \in \mathcal {K}, t\in \mathcal {T}, \omega \in \Omega }{} \end{aligned}$$

(4j)

$$\begin{aligned}&{V_{ikt}+ \sum \limits _{{\scriptscriptstyle j\in \mathcal {L} \cup \mathcal {S}, j \ne i}}f^{\omega }_{ijkt}}{= V_{ik(t-1)}+ g^{\omega }_{ikt} + h^{\omega }_{ikt}} \nonumber \\&\quad + \sum \limits _{{\scriptscriptstyle j\in \mathcal {L} \cup \mathcal {S}, j \ne i}}f^{\omega }_{jikt}, \quad \forall i \in \mathcal {L}, k \in \mathcal {K}, t\in \mathcal {T}{\setminus }\{0\}, \omega \in \Omega \end{aligned}$$

(4k)

$$\begin{aligned}&{U_{ikt}}{= (1 - \alpha _{ikt})(U_{ik(t-1)}+1), \quad \forall i \in \mathcal {S}, k \in \mathcal {K}, t\in \mathcal {T}{\setminus }\{0\}}{} \end{aligned}$$

(4l)

$$\begin{aligned}&{V_{ikt}}{\le \alpha _{ikt} \phi _{ik}, \quad \forall i \in \mathcal {S}, k \in \mathcal {K}, t\in \mathcal {T}{\setminus }\{0\}}{} \end{aligned}$$

(4m)

$$\begin{aligned}&{\sum \limits _{{\scriptscriptstyle j\in \mathcal {L} \cup \mathcal {S}, j \ne i }}f^{\omega }_{ijkt}}{\le V_{ikt}, \quad \forall i \in \mathcal {S}, k \in \mathcal {K}, t\in \mathcal {T}{\setminus }\{0\}, \omega \in \Omega }{} \end{aligned}$$

(4n)

$$\begin{aligned}&{\sum \limits _{{\scriptscriptstyle j\in \mathcal {L} \cup \mathcal {S}, j \ne i }}f^{\omega }_{jikt}}{\le V_{ikt}, \quad \forall i \in \mathcal {S}, k \in \mathcal {K}, t\in \mathcal {T}{\setminus }\{0\}, \omega \in \Omega }{} \end{aligned}$$

(4o)

$$\begin{aligned}&{h^{\omega }_{ikt}}{\le V_{ikt}, \quad \forall i \in \mathcal {S}, k \in \mathcal {K}, t\in \mathcal {T}{\setminus }\{0\}, \omega \in \Omega }{} \end{aligned}$$

(4p)

$$\begin{aligned}&{V_{ikt} + \sum \limits _{{\scriptscriptstyle j\in \mathcal {L} \cup \mathcal {S} \cup \{n\}, j \ne i}}f^{\omega }_{ijkt}}{=V_{ik(t-1)}- \alpha _{ikt} D^{\omega }_{ikt} + h^{\omega }_{ikt}} \nonumber \\&\quad + \sum \limits _{{\scriptscriptstyle j\in \mathcal {L} \cup \mathcal {S}, j \ne i}}f^{\omega }_{jikt}, \quad \forall i \in \mathcal {S}, k \in \mathcal {K}, t\in \mathcal {T}{\setminus }\{0\}, \omega \in \Omega \end{aligned}$$

(4q)

$$\begin{aligned}&{x_{it},y_{it}}{\in \{0,1\}, \quad \forall i\in \mathcal {L}, t\in \mathcal {T}}{} \end{aligned}$$

(4r)

$$\begin{aligned}&{\alpha _{ikt}}{\in \{0,1\}, \quad \forall i \in \mathcal {S}, k \in \mathcal {K}, t\in \mathcal {T}{\setminus }\{0\}}{} \end{aligned}$$

(4s)

$$\begin{aligned}&{U_{ikt}}{\in \mathbb {Z^+}, \quad \forall i \in \mathcal {S}, k \in \mathcal {K}, t\in \mathcal {T}}{} \end{aligned}$$

(4t)

$$\begin{aligned}&{V_{ikt}}{\ge 0, \quad \forall i \in \mathcal {L} \cup S, k \in \mathcal {K}, t\in \mathcal {T}}{} \end{aligned}$$

(4u)

$$\begin{aligned}&{g^{\omega }_{ikt}, h^{\omega }_{ikt}}{\ge 0, \quad \forall i \in \mathcal {L}, k \in \mathcal {K}, t\in \mathcal {T}, \omega \in \Omega }{} \end{aligned}$$

(4v)

$$\begin{aligned}&{f^{\omega }_{ijkt}}\ge 0, \quad \forall i, j\in \mathcal {L}\cup \mathcal {S} \cup \{n\}, k \in \mathcal {K},\nonumber \\&t\in \mathcal {T}, \omega \in \Omega \end{aligned}$$

(4w)